Convergence of tamed Euler schemes for a class of stochastic evolution equations

István Gyöngy,Sotirios Sabanis,David Šiška

doi:10.1007/s40072-015-0057-7

Abstract

We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done by using the recently developed tamed Euler method, which employs a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization.

Highlights

In this paper we investigate the convergence of full discretizations, explicit in time, of stochastic evolution equations du(t) = Au(t)dt + Bu(t)dW (t), t ∈ [0, T ]
If the operator appearing in the drift term grows faster than linearly one would, in general, not expect the explicit Euler scheme to be convergent
Hutzenthaler, Jentzen and Kloeden [9] have observed that the absolute moments of explicit Euler approximations for stochastic differential equations with super-linearly growing coefficients may diverge to infinity at finite time

Summary

Introduction

Hutzenthaler, Jentzen and Kloeden [9] have observed that the absolute moments of explicit Euler approximations for stochastic differential equations with super-linearly growing coefficients may diverge to infinity at finite time. This led to development of “tamed” Euler schemes for stochastic differential equations. Hutzenthaler, Jentzen and Kloeden [11] have demonstrated that to apply multilevel Monte Carlo methods (see Heinrich [7,8] and Giles [5]) to stochastic differential equations with super-linearly growing coefficients one must “tame” the explicit Euler scheme.

Main results

A priori estimates

Convergence

Examples